1. 2.3 Real and Non Real Roots of a Polynomial Polynomial Identities Secondary Math 3

2. WARM UP Given the polynomial 1. What is the degree? 2. What is the end behavior? 3. What are the zeros? 4. Sketch a graph.

3. Complex Zeros The Fundamental Theorem of Algebra states that a polynomial of nth degree will have n complex zeros. Complex zeros (a + bi) can be either real numbers or non real (imaginary) numbers. When roots of a polynomial include non real numbers (the graph will not cross the x-axis) it will always include a non real number and its conjugate (a + bi and a – bi). Thus, non real roots come in pairs.

4. Three polynomials of degree 4 are graphed below. Describe the roots. r(x) – has four real roots q(x) – has two real roots and two non real roots p(x) – has four non real roots

5. Three polynomials of degree 3 are graphed below. Describe the roots. r(x) – one real root and two non real roots q(x) – has three real roots p(x) – has three real roots

6. Questions A polynomial has a degree of 8. Which of the following could be the number of real roots? 2, 3, 4, 7, or 8 A polynomial with real coefficients has a degree of 5. Which of the following could be the number of complex non-real roots? 2, 3, 4, 6, or 7 Assume that the degree of a polynomial is odd. The number of real roots would be even or odd?

7. Polynomial Identities Perfect Square TrinomialDifference of Squares

8. Polynomial Identities Cubic Polynomials

9. Polynomial Identities Sum of CubesDifference of Cubes

10. Polynomial Identities Trinomial Leading Coefficient of 1Sum of Squares

11. Proving Polynomial Identities To prove an identity you simplify or change one side to get the other side.

12. Quadratic Formula Proof Solve by completing the square.

13. Irreducible or Prime Polynomial A polynomial with integer coefficients that can not be factored into polynomials of lower degree, also with integer coefficients.

14. Examples – Multiply using Polynomial Identities

16. Examples – Factor expressions using the Polynomial Identities.